An important example of this is the momentum flux tensor. If ρ is the density and
v is the velocity,
ρvi is the
ith component in the direction
Oi . The rate at which this momentum crosses a unit area normal to
Oj is the tensor,
ρvivj.

Scalar multiplication and addition

If α is a scalar and
A a second order tensor, the scalar product of α and
A is a tensor α
A each of whose components is α times the corresponding component of
A .

The sum of two second order tensors is a second order tensor each of whose components is the sum of the corresponding components of the two tensors. Thus the
ijth component of
A +
B is
Aij +
Bij . Notice that the tensors must be of the same order to be added; a vector can not be added to a second order tensor. A linear combination of tensors results from using both scalar multiplication and addition. α
A + ß
B is the tensor whose
ijth component is α
Aij + ß
Bij . Subtraction may therefore be defined by putting
α=1 ,
ß=-1 .

Any second order tensor can be represented as the sum of a symmetric part and an antisymmetric part. For

represents
A as the sum of a symmetric tensor and antisymmetric tensor.

Contraction and multiplication

As in vector operations, summation over repeated indices is understood with tensor operations. The operation of identifying two indices of a tensor and so summing on them is known as
contraction .
Aii is the only contraction of
Aij ,

and this is no longer a tensor of the second order but a scalar, or a tensor of order zero. The scalar
Aii is known as the trace of the second order tensor
A . The notation tr
A is sometimes used. The contraction operation in computing the trace of a tensor
A is analogous to the operation in the calculation the magnitude of a vector
a , i.e.,
|a|2=a⋅a=a1a1+a2a2+a3a3

If
A and
B are two second order tensors, we can form 81 numbers from the products of the 9 components of each. The full set of these products is a fourth order tensor. Contracted products result in second order or zero order tensors. We will not have an occasion to use products of tensors in our course.

The product
Aijaj of a tensor
A and a vector
a is a vector whose
ith component is
Aijaj . Another possible product of these two is
AijaI . These may be written
A ⋅
a and
a ⋅
A , respectively. For example, the diffusive flux of a quantity is computed as the contracted product of the transport coefficient tensor and the potential gradient vector, e.g.,
q=-k⋅∇T

The vector of an antisymmetric tensor

We showed earlier that a second order tensor can be represented as the sum of a symmetric part and an antisymmetric part. Also, an antisymmetric tensor is characterized by three numbers. We will later show that the antisymmetric part of the velocity gradient tensor represents the local rotation of the fluid or body. Here, we will develop the relation between the angular velocity vector,
ω , introduced earlier and the corresponding antisymmetric tensor.

Recall that the relative velocity between a pair of points in a rigid body was described as follows.

The notation
vecΩ is sometimes used for
ω . In summary, an antisymmetric tensor is completely characterized by the vector,
vecΩ .

Canonical form of a symmetric tensor

We showed earlier that any second order tensor can be represented as a sum of a symmetric part and an antisymmetric part. The symmetric part is determined by 6 numbers. We now seek the properties of the symmetric part. A theorem in linear algebra states that a symmetric matrix with real elements can be transformed by its eigenvectors to a diagonal matrix with real elements corresponding the eigenvalues. (see Appendix A of Aris.) If the eigenvalues are distinct, then the eigenvector directions are orthogonal. The eigenvectors determine a coordinate system such that the contracted product of the tensor with unit vectors along the coordinate axis is a parallel vector with a magnitude equal to the corresponding eigenvalue. The surface described by the contracted product of all unit vectors in this transformed coordinate system is an ellipsoid with axes corresponding to the coordinate directions.

The eigenvalues and the scalar invariants of a second order tensor can be determined from the characteristic equation.

Assignment 2.1

Relative velocity of points in a rigid body. If
x and
y are two points inside a rigid body that is translating and rotating, determine the relation between the relative velocity of these two points as a function of their relative positions. If
x and
y are points on a line parallel to the axis of rotation, what is their relative velocity? If
x and
y are points on opposite sides of the axis of rotation but with equal distance, r, what is their relative velocity? Draw diagrams.

Prove that:
a •(
b ×
c ) = (
a ×
b )•
c

Show
a •(
b ×
c ) vanishes identically if two of the three vectors are proportional of one another.

fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.

Tarell

what is the actual application of fullerenes nowadays?

Damian

That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.

Tarell

Join the discussion...

what is the Synthesis, properties,and applications of carbon nano chemistry

Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.